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A308267 Numbers which divide their Zeckendorffian format exactly. 0
1, 2, 4, 7, 8, 14, 16, 28, 31, 32, 56, 62, 64, 83, 112, 124, 127, 128, 166, 224, 248, 254, 256, 332, 397, 448, 496, 508, 511, 512, 664, 794, 891, 896, 992, 1016, 1022, 1024, 1163, 1328, 1588, 1782, 1792, 1984, 2032, 2044, 2047, 2048, 2326, 2656, 3176, 3441, 3564, 3584, 3968 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Indices of A048678 which divide into their Zeckendorffian format.

Curiously seems to include the powers of 2, some Mersennes and perfects.

Fixed points of the Zeckendorf function are 2^k.

All numbers 2^(2k+1)-1 are in this sequence, and that if n is in this sequence then so is 2n. - Charlie Neder, May 17 2019

LINKS

Table of n, a(n) for n=1..55.

EXAMPLE

The first few terms of A048678 are 1,2,5,4,9,10,21,8. 2 is a multiple of 2, 5 isn't a multiple of 3, 4 is a multiple of 4, 9 isn't a multiple of 5, 10 isn't a multiple of 6, 21 is a multiple of 7, etc.

MATHEMATICA

Select[Range[4000], Divisible[FromDigits[Flatten[IntegerDigits[#, 2] /. {1 -> {0, 1}}], 2], #] &] (* Amiram Eldar, Jul 08 2019 after Robert G. Wilson v at A048678 *)

PROG

(Haskell)

import Data.Numbers.Primes

bintodec :: [Int] -> Int

bintodec = sum . zipWith (*) (iterate (*2) 1) . reverse

decomp :: (Integer, [Integer]) -> (Integer, [Integer])

decomp (x, ys) = if even x then (x `div` 2, 0:ys) else (x - 1, 1:ys)

zeck :: Integer -> String

zeck n = bintodec (1 : snd (last $ takeWhile (\(x, ys) -> x > 0) $ iterate decomp (n, [])))

output :: [Integer]

output = filter (\x -> 0 == zeck x `mod` x) [1..100]

CROSSREFS

Cf. A048678, A083420 (subsequence).

Sequence in context: A324588 A201364 A225318 * A177805 A003591 A029746

Adjacent sequences:  A308264 A308265 A308266 * A308268 A308269 A308270

KEYWORD

nonn,base,new

AUTHOR

Dan Dart, May 17 2019

STATUS

approved

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Last modified July 15 20:24 EDT 2019. Contains 325056 sequences. (Running on oeis4.)